Calibration of tri-axial MEMS accelerometers in the low ... · a capacitive MEMS accelerometer. A low-frequency range of vibration has been investigated, 3 and 6Hz, with amplitude

J. Sens. Sens. Syst., 7, 245–257, 2018https://doi.org/10.5194/jsss-7-245-2018© Author(s) 2018. This work is distributed underthe Creative Commons Attribution 4.0 License.

Calibration of tri-axial MEMS accelerometers in thelow-frequency range – Part 1: comparison

among methods

Giulio D’Emilia1, Antonella Gaspari1, Fabrizio Mazzoleni2, Emanuela Natale1, and Alessandro Schiavi21Department of Industrial and Information Engineering and of Economics, University of L’Aquila,

L’Aquila, 67100, Italy2INRiM - National Institute of Metrological Research, Turin, 10135, Italy

Correspondence: Giulio D’Emilia ([email protected])

Received: 8 February 2018 – Revised: 26 March 2018 – Accepted: 28 March 2018 – Published: 6 April 2018

Abstract. Two alternative experimental procedures for the calibration of tri-axial accelerometers have beencompared with traditional methods, performed according the procedures stated in the standard ISO 16063-21.Standard calibration is carried out by comparison with a laser Doppler vibrometer (LDV), used as a primaryreference transducer. The main sensitivities have been investigated and, where applicable, also transverse ones.Many aspects have been evaluated: the hypotheses about transverse sensitivities, the simplicity of the procedure,the number of measurements needed, and the effect of typology of transducer, depending on electrical andgeometrical contributions. Two different accelerometers have been tested, a piezo-electric accelerometer anda capacitive MEMS accelerometer. A low-frequency range of vibration has been investigated, 3 and 6 Hz, withamplitude of acceleration ranging from 2 to 20 ms−2. A satisfactory reproducibility of methods has been verified,with percentage differences less than 2.5 %. Anyway, pros and cons of each method are also discussed withreference to their possible use for easy and quick calibration of low-cost tri-axial accelerometers.

1 Introduction

In recent years, there has been a growing interest in micro-electro-mechanical system accelerometers (MEMS), due totheir low cost, the possibility of embedding these deviceswithin wireless sensor networks, and the capability of detect-ing low-amplitude and low-frequency vibrations, operationswhich are not always feasible with the conventional low-costsensor boards (Batista et al., 2011; Sabato et al., 2017). Itshould be considered that MEMS accelerometers, in com-parison to high-performance piezo-electric transducers, ex-hibit lower accuracy in consumer grade applications; how-ever, in the context of extensive applications, such as largesensor networks, in which high accuracy on a wide rangeof frequency and amplitude is not needed, the technical per-formances of these accelerometers are considered adequate(Schiavi et al., 2015).

Low-frequency vibration measurements are of great in-terest in many different fields such as, for instance, energy

production (Ripper et al., 2017), structural health monitor-ing (SHM) of buildings and of civil infrastructures (Sabatoet al., 2017; Ranieri et al., 2013), and geotechnical applica-tions (Czech and Gosk, 2017) in the field of human vibra-tion and bio-dynamics (Griffin, 2014), mainly because of theincreasing development of MEMS embedded in mobile de-vices (Halim and Park, 2013) and in applications in the fieldof the Internet of Things (IoT) (Borgia, 2014).

The evaluation of sensitivities is a key point of the vi-bration measurements for MEMS accelerometers, and it re-quires adequate attention from the design phase up to the re-alization and the use of this kind of sensor.

In this paper, new calibration techniques are investigated,with the purpose of increasing the flexibility of the traditionalcalibration procedures (ISO 16063-1, 1998; ISO 16063-11,1999; ISO 16063-31, 2009), to tailor them to the variableconditions and requirements of the field while guaranteeing,at the same time, traceability and accuracy (Schiavi et al.,2015; D’Emilia et al., 2011, 2015). An example is the pos-

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246 G. D’Emilia et al.: Calibration of tri-axial MEMS accelerometers

sibility of developing procedures that could be implementedboth on-line and in-line (Frosio et al., 2009, 2012; Glueck etal., 2012; Fong et al., 2008).

On-line calibration refers to the possibility of calibratingduring assembly in industrial processes, for continuous mon-itoring and controlling of processes. In-line calibration refersto the calibration of sensors installed on an industrial plant,directly carried out on the production line to be monitoredwithout moving it to a laboratory.

The possibility of calibrating sensors in the field is an in-teresting issue not only for practical and economic reasons,but also from a technical point of view. In fact, the MEMSsensor output depends on temperature and, in general, onother environmental conditions (Wu et al., 2002). Therefore,these accelerometers have to be calibrated in the field whena realistic assessment of the measurement uncertainty is animportant requirement (Frosio et al., 2009, 2012; Glueck etal., 2012; Fong et al., 2008).

The research for new methods for the on-line and in-line calibration of MEMS is confirmed as one of the pointsof main relevance. In fact, some procedures are proposedbased on different algorithms. In Chen and Han (2011) thewavelet neural network is used for optimizing and compen-sating for the variation of the MEMS acceleration sensorsdue to temperature change. Rohac et al. (2015) propose amethod for calibration of MEMS tri-axial inertial sensors,using a gravity-based calibration method under static condi-tions. Geist et al. (2017) propose a methodology based onthe linearization, used for the reduction of the measurementerror of the device. The optimization algorithm is validatedon an experimental set-up, considering the accelerometer instatic state, and rotating it randomly in 30 different orien-tations. These methods are static-based methods and do notconsider aspects arising from the experimental practice (e.g.the dynamic behaviour of the phenomena analysed).

In summary, the following main requirements for testingand calibration of accelerometers for the above-mentionedapplications should be met:

– to lower the cost of calibration and testing (low-cost cal-ibration);

– to enhance the sensors’ operability, also in dynamicconditions, similar to those of their actual use (dynamicconditions);

– to guarantee the traceability of the methodology, up tothe primary calibration (traceability);

– to streamline the production process and to reduce lossin times and costs of possible deviations from the vali-dated information, also in the case of on-line and in-linecalibration (on-line and in-line calibration);

– to avoid complex models and algorithms, in order to beeasily transferred onto the field, thus increasing their op-erability (simplicity and operability).

A quantitative comparison among three methods is carriedout. Two methods have been selected that are potentiallyable to fulfil the above-mentioned requirements, in a differ-ent way, with respect to the standard method, used as a refer-ence. The standard method, Method 1, is performed accord-ing to the procedures stated in the ISO Standard 16063 se-ries (ISO 16063-1, 1998; ISO 16063-11, 1999; ISO 16063-21, 2003; ISO 16063-31, 2009), to support the traceability.The second one, Method 2 (Schiavi et al., 2015), allows us todetermine the three main sensitivities, simultaneously. Thelast one (Method 3; D’Emilia et al. ,2015, 2016a, b) givesthe possibility of obtaining the main sensitivities, transversesensitivities and offset terms.

In Sect. 2, the methods are described and a discussionof the way each method fits the above requirements is car-ried out. In Sect. 3, the test bench used for the experimentalset-up, together with the typology of the accelerometers thathave been used, and the planned tests are outlined. The indi-cators used for the comparison purposes are also identified.In Sect. 4 the results for the two typologies of sensors anal-ysed are presented and discussed. Conclusions and hints forfuture works end the paper.

2 Calibration methods

In the following sections, the methods of calibration and theparameters considered for the comparison will be shown.Then, the test bench used for the experimental tests will alsobe described.

2.1 Method 1: the standard ISO 16063 series

The standard (ISO 16063-1, 1998) defines the sensitivity fora linear transducer as the ratio of the input during sinusoidalexcitation, parallel to a specified axis of sensitivity at themounting surface.

The procedure to be used for primary calibration of rec-tilinear accelerometers to obtain magnitude and phase lagof the complex sensitivity by steady-state sinusoidal vibra-tion and laser interferometry is described in ISO 16063-11 (1999). The sinusoidal motion applied by the vibrationgenerator is along a well-defined straight line, with negligi-ble lateral motions.

In this paper, the traceability to primary national standardsthrough a secondary standard is accomplished by applyingthe specifications in ISO 16063-21 (2003). Additionally, thestandard (ISO 16063-31, 2009) defines the transverse sensi-tivity of an accelerometer, ST, as the sensitivity to acceler-ation applied at right angles to its geometric axis. ST, cor-responding to a specific test direction, is calculated as (1),where Vout is the amplitude of the output signal of the trans-ducer and aT is the amplitude of the acceleration in the testdirection. The test direction is perpendicular to the axis of

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G. D’Emilia et al.: Calibration of tri-axial MEMS accelerometers 247

sensitivity.

ST =Vout

aT(1)

The standard ISO 16063-31 (2009) deals with uni-axial ac-celerometers, excited along directions perpendicular to theirsensitive axes.

Extending the indications of the standard to the case ofa tri-axial accelerometer, each axis of the sensor could betested by applying a known acceleration to another axis, withthe purpose of calculating according to Eq. (1) the transversesensitivity of the first axis when excited along the directionof the second one.

Then, by exciting each single axis of a sensor, it is possi-ble to calculate six transverse sensitivities, which express theeffect of the acceleration along a single axis, with respect tothe other ones.

These transverse sensitivities will be indicated as Sxy , Sxz,Syx , Syz, Szx , and Szy , where the first subscript indicates theaxis with reference to which the sensitivity is calculated, andthe second one indicates the direction of the excitation.

A similar approach is applied for the determination of themagnitude of the acceleration sensitivities (Sxx , Syy , Szz).The phase lag is assumed to be negligible.

As an example, when the vibration is according to thex axis, the sensitivity is calculated as Eq. (2), while the trans-verse sensitivities Syx and Szx can be evaluated as Eqs. (3)and (4), where Vx , Vy and Vz are the amplitudes of the x, yand z outputs of the accelerometer under test, and ax is theamplitude of the reference signal in the x direction.

Sxx =Vx

ax(2)

Syx =Vy

ax(3)

Szx =Vz

ax(4)

The same procedure is used for the other sensitivities, whenthe vibration is along the y axis (Syy , Sxy , Szy) and z axis(Szz, Sxz, Syz), respectively.

2.2 Method 2

Method 2 involves the simultaneous excitation of the threeaxes of the accelerometer under test.

For this purpose, the accelerometer has to be mounted ontothe surface of a clamp, inclined at an angle θ = 35◦ with re-spect to the horizontal plane (Fig. 1) on which the motionis realized; furthermore, the accelerometer has to be rotatedon the clamp surface with an angle α = 45◦, in order to si-multaneously excite the three axes in the same way, with asingle horizontal sinusoidal acceleration (Fig. 1) (Schiavi etal., 2015).

Figure 1. Inclined clamp – scheme.

In this way, accelerations of similar amplitude are realizedalong the three axes, which can be simultaneously calibrated.

The reference accelerations along the three axes can beobtained as follows in Eqs. (5), (6) and (7):

ax =−aref · cos(θ ) · sin(α), (5)ay =−aref · cos(θ ) · cos(α), (6)az = aref · sin(θ ), (7)

where aref is the acceleration in the motion direction, mea-sured by a reference sensor.

The output signal and the reference one are analysed bythe fast Fourier transform in correspondence to the oscilla-tion frequency, with the purpose of evaluating their spectralamplitudes. The constant terms, gravity-dependent, do notaffect the results.Sxx is evaluated according to Eq. (2), where Vx is the spec-

tral amplitude of the output signal of the x axis of the ac-celerometer under test, and ax is the spectral amplitude ofthe reference signal in the same direction.

The same approach allows us to evaluate Syy and Szz, re-spectively, when the other measuring axes of the accelerom-eter are considered.

This approach does not allow us to calculate the transversesensitivities, since similar acceleration components are re-alized along all axes, simultaneously, without the possibil-ity of extracting the effect of transverse sensitivities. A peri-odic verification that the transverse sensitivities are negligi-ble (< 5 % with respect to the main sensitivities) is necessaryfor the typology of sensor considered, according to Method1, described in Sect. 2.1.

2.3 Method 3

This method is based on the following model, describing therelation between the input accelerations and the output sig-nals (Eq. 8), where V = (Vi) is the output array, A= (ai) isthe reference accelerations array, S= (Sij ) is the sensitivitymatrix and Q= (qi) represents the offset array. VxVyVz

= Sxx Sxy SxzSyx Syy SyzSzx Szy Szz

· axayaz

+ qxqyqz

(8)

Then, a total of 12 parameters (main sensitivities, transversesensitivities and offsets) are evaluated; a linear least squares

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Table 1. Pros and cons of the methods with respect to requirements.

Requirements Method 1 Method 2 Method 3

Low-costcalibration

No Yes Yes

Dynamicconditions

Suitable for low-frequencyvibration

Suitable for low-frequency vibration Suitable for low-frequency vibration

Traceability Close similarity to the stan-dard method

After comparison with Method 1 After comparison with Method 1

On-line and in-linecalibration

Analysis based on the fre-quency amplitude evalua-tion

Analysis based on the frequency am-plitude evaluation

High statistical robustness based on apoint-by-point comparison. More an-gular positions are required, but theycould be anything. The vibration mo-tion law can be set according to thespecific application (not necessary sinu-soidal).

Simplicity andoperability

Possibility of measur-ing (main) sensitivities,transverse sensitivities.Working with a sinusoidalmotion law parallel toeach measuring axis underexamination. No effect oftransverse sensitivity on thecalculation of sensitivity.

Possibility of simultaneous measur-ing of the (main) sensitivities. Themeasuring axes do not correspondto the motion direction. Fixed an-gular positioning. Working with asinusoidal motion law. The effectof transverse sensitivities cannot beevaluated.

Possibility of measuring (main) sensi-tivities, transverse sensitivities and off-set. The measuring axes do not corre-spond to the motion direction. More an-gular positions are required, but theycould be whatever. The vibration mo-tion law can be set according to thespecific application (not necessary sinu-soidal).

optimization is used to estimate them, as described in thereferences (D’Emilia et al., 2015, 2016a, b).

To avoid dependence among the input data for the leastsquares optimization, the sensor has to be positioned in dif-ferent angular positions with respect to the motion direction.The sensor should be mounted with an angle θ with respectto the horizontal plane on which the motion is carried out,repeating measurements for different angles α (Fig. 1). InMethod 3, angle θ could be anything. In this comparison, itwill be set the same as Method 2 (θ = 35◦).

The method requires a minimum number of four differ-ent input acceleration vectors, although more measurementsmay give a more robust calibration. The inputs correspond todifferent values for time and angles α.

The reference accelerations along the three axes can beobtained from Eqs. (9), (10), and (11), where g is the gravityacceleration, and aref(t) is the time-varying acceleration inthe motion direction and measured by a reference sensor.

ax(t)=−aref(t) sin(α)cos(θ )− g · sin(θ ) · sin(α) (9)ay(t)=−aref(t) · cos(α) · cos(θ )− g · sin(θ ) · cos(α) (10)az(t)= aref(t) · sin(θ )− g · cos(θ ) (11)

It must be pointed out that the constant terms in formu-las (12–14), due to the components of the gravity accelera-tion, should not be considered for sensors, like piezo-electricones, that are not sensitive to constant accelerations.

Figure 2. Time behaviour of accelerometer outputs and referenceinputs: example.

In Fig. 2, as an example, the time behaviour of both theaccelerometer outputs (Vx , Vy , Vz) and the reference signals(ax , ay , az) is shown, in the time domain (α = 30◦, sam-pling frequency 1000 Hz, excitation frequency f1 = 3 Hz,and number of inputs= 333, corresponding to one period ofoscillation).



Figure 3. APS 113 ELECTRO-SEIS horizontal vibrating table.

2.4 Discussion of the methods

Table 1 summarizes some considerations concerning the spe-cific characteristics of the methods to be compared, in orderto give information about the possibility of using them in-field, for the on-line and in-line calibration of MEMS, andabout their capability of satisfying requirements involved inthese applications.

2.5 Test bench and test procedure

For the comparison among calibration methods, two differentaccelerometers have been considered:

– a MEMS accelerometer (Sequoia FastTracer®) with acapacitive transduction system, with digital output to acomputer, via a standard USB port. The output indicatesthe acceleration (nominal sensitivity of the order of 1,dimensionless);

– a piezo-electric accelerometer (PCB 356A15) withanalogue output (nominal sensitivity of the order of10 mV m−1 s2).

The test bench used is a vibrating table with a horizontallinear slide, the APS 113 ELECTRO-SEIS shaker (Fig. 3).It is a long-stroke, electro-dynamic force generator specifi-cally suitable for low-frequency vibration testing. The slideis moved according to a sinusoidal law.

In Method 1 the accelerometer is fixed directly on the hor-izontal vibrating table, with one of the three axes parallel tothe motion direction. All three axes are tested in this way,recursively.

In Method 2 and Method 3 an inclined steel clamp is fixedon the vibrating table; the inclination angle θ is 35◦ withrespect to the horizontal plane (Fig. 4). The inclined steelclamp allows us to obtain a specified angle α (Method 2) ordifferent angles α (Method 3). The motion of the vibratingtable is accurately monitored by a laser Doppler vibrome-ter (LDV), as depicted in Fig. 4c.

The amplitude of the reference acceleration signal is ob-tained by applying Eq. (12), ω being the pulsation of the si-nusoidal motion and vvib the velocity measured by the LDV.

aref = ω · vvib (12)

The data acquisition system (DAQ) used is the NI USB-4431by National Instruments. The module consists of a singleanalogue output and four analogue input channels for read-ing (one is connected to the LDV and the other three to theoutputs of the PCB accelerometer under test); each channel isequipped with antialiasing filters. The output channel drivesthe vibrating table. LabVIEW software is used for DAQ sig-nal acquisition.

2.6 Description of the experiments

Tests are carried out at the excitation frequencies of f1 =

3 Hz and f2 = 6 Hz, at different amplitudes in the range 2 to20 ms−2. Independent tests are carried out in two differentconfigurations.

– Method 1: parallel excitation, with respect to the mea-suring axes of the accelerometer. The sensors are ex-cited along the main measuring components x, y andz axes, according to standards (ISO 16063-1, 1998; ISO16063-11, 1999; ISO 16063-31, 2009).

– Method 2: inclined excitation with respect to the mea-suring axes of the accelerometer (θ = 35◦, α= 135◦).

– Method 3: inclined excitation, with respect to the mea-suring axes of the accelerometer. Each sensor is rotated,according to four different angles, α, between the x axisof the accelerometer and the horizontal. Depending onthe specific sensor under test, α takes the following val-ues: 0, 30, 90, 120, 135, and 210◦, as in Fig. 5.

2.7 Comparison between post-processing techniques

In order to determine whether and to what extent the cali-bration methodologies are equivalent, the following resultsare compared, obtained through the application of the threeabove-mentioned methodologies:

– calibration tests of the same accelerometer, on the sametest bench (high-performance linear slide);

– calibration tests of different accelerometers, of differenttechnology and quality levels.

The comparison of the results and the assessment of the pos-sible equivalence between methods is made by the param-eters whose symbols are defined in Table 2, where M andN assume the following numbers: 1, 2, 3, depending on thespecific method applied (e.g. d_rel(Sxx)MN , for M = 3 andN = 2 stands for d_rel(Sxx)32, i.e. the relative difference be-tween Method 3 and Method 2).



Figure 4. Particulars of the test bench: indication of the directions of acceleration. Piezo-electric (a) and MEMS (b) sensors, mounted onthe inclined clamp. Laser vibrometer (c).

Table 2. Parameters used for the comparison.

Quantity Parameter Symbol

Main sensitivity,offset

Relative standard deviation (s_rel) s_rel (Sxx)s_rel

(Syy

)s_rel(Szz)s_rel(qx)s_rel

(qy)

s_rel(qz)

Main sensitivity Relative difference betweenmethods (d_rel)

d_rel (Sxx)MNd_rel

(Syy

)MNd_rel (Szz)MN

Transverse sensitivity Relative transverse sensitivity (rel_S) rel_Sxy, rel_Sxz,rel_Syx, rel_Syz,rel_Szxrel_Szy

Details on the evaluation of each parameter are reported inAppendix A, taking as an example the main sensitivity Sxx(i.e. the main sensitivity along the measuring x axis of theaccelerometer).

In the following, the quantities used for comparison aresummarized:

– the sensitivities and transverse sensitivities of each ac-celerometer and the related variability;

– the relative differences between methods of the mainand the transverse sensitivities;

– sensitivities obtained by applying each method, takinginto account the specific uncertainty.

3 Results

This section is organized as follows: Sects. 4.1 and 4.2 re-port the results obtained for the two accelerometers under test

(piezo-electric and MEMS), respectively. In each of them,the results of the three methods described are shown and acomparison among them is carried out.

For the results obtained for the sensitivity matrix, a graph-ical representation in the 3-D space is used according to thescheme of Fig. 6. Each point represents a row of the sensitiv-ity matrix. The sensitivity components are expressed in mil-livolt per metre second squared (mV m−1 s2), for the piezo-electric accelerometer, while they are dimensionless for theMEMS accelerometer, since the digital output values are di-rectly expressed in metres per second squared (m s−2).

The x axis represents the locus of the ideal response alongthe x axis itself of the three-axis accelerometer (Sxx 6= 0;Sxy = 0; Sxz= 0), meaning negligible transverse sensitivi-ties. The same applies to the y axis and z axis.



Figure 5. Angles of rotation of the accelerometers: (a) MEMS accelerometer; (b) piezo-electric accelerometer.

Table 3. Piezo-electric accelerometer: comparison between methods (%).

Frequency Methods d_rel (Sxx)MN d_rel(Syy

)MN d_rel (Szz)MN

M = 2; N = 1 0.51 1.3 2.03 Hz M = 3; N = 1 −0.18 −0.24 0.91

M = 3; N = 2 −0.69 −1.5 −1.1

M = 2; N = 1 0.54 1.2 1.96 Hz M = 3; N = 1 −0.20 −1.1 −1.3

M = 3; N = 2 −0.74 −2.3 −3.2

Figure 6. 3-D diagram for the sensitivity matrix: scheme and nota-tion.

3.1 Piezo-electric accelerometer

3.1.1 Method 1

Figure 7 represents the sensitivity matrix obtained by ap-plying Method 1, as described in Sect. 2.1, at 3 Hz (upperside) and 6 Hz (bottom side). The relative standard deviation(s_rel), defined in Table 1 and Appendix A, of all main sen-sitivities is negligible, being less than 0.03 %.

Figure 7. Piezo-electric accelerometer: sensitivity matrix obtainedby means of Method 1 (3-D diagram).

3.1.2 Method 2

Figure 8 represents the sensitivity matrix obtained by ap-plying Method 2, as described in Sect. 2.2, at 3 Hz (upperside) and 6 Hz (bottom side). The relative standard deviation(s_rel) of all main sensitivities is negligible, being less than0.04 %.



Table 4. MEMS accelerometer: comparison between methods (%).

Frequency Methods d_rel (Sxx)MN d_rel(Syy

)MN d_rel (Szz)MN

M = 2; N = 1 0.73 −0.12 −0.193 Hz M = 3; N = 1 2.8 −0.49 1.1

M = 3; N = 2 2.1 −0.37 −0.93

M = 2; N = 1 0.68 −0.25 0.116 Hz M = 3; N = 1 1.8 1.5 1.4

M = 3; N = 2 1.1 1.7 1.3



3.1.3 Method 3

Figure 9 represents the sensitivity matrix obtained by ap-plying Method 3, as described in Sect. 2.3, at 3 Hz (upperside) and 6 Hz (bottom side). The relative standard devia-tion (s_rel) of all main sensitivities is negligible (less than0.08 %). As a first approximation, the mean main sensitiv-ities appear coherent to those obtained through Method 1(Fig. 7). The offset vectors at f1 = 3 Hz and f2 = 6 Hz are asin the following, and they are considered negligible; there-fore, their variability is meaningless. For the piezo-electricaccelerometer under investigation, the output values are inmillivolt (mV), and as a consequence the offset values are

Figure 10. Piezo-electric accelerometer: comparison betweenmethods (main sensitivity along the x axis, Sxx ).

expressed in terms of voltage. qxqyqz

f 1

=

0.3113mV0.5427mV0.06098mV

; qxqyqz

f 2

=

0.2477mV0.5668mV0.5839mV

.3.1.4 Comparison between methods

Relative differences between methods (d_rel) obtained byapplying the three methods are reported in Table 3, when thepiezo-electric sensor is excited at 3 and 6 Hz, respectively.

Differences between Method 2 and Method 1 are al-ways positive up to 2 %. Differences between Method 3 andMethod 1 are both negative and positive and in the range±1.3 %. The highest differences arise between Method 3 andMethod 2, being d_rel(Szz)32 =−3.2 % (at 6 Hz).

Figures 10–12 highlight, for each axis, the relative posi-tion and differences among methods. Expanded uncertainty(k= 2) of results, estimated according to D’Emilia et al.(2018), is also represented in Figs. 10–12 by error bars.

3.1.5 Reproducibility: transverse sensitivities

The relative transverse sensitivities (rel_S) for all axes, asdefined in Table 1 and Appendix A, are all under 3 %. These



Figure 11. Piezo-electric accelerometer: comparison betweenmethods (main sensitivity along the y axis, Syy ).

Figure 12. Piezo-electric accelerometer: comparison betweenmethods (main sensitivity along the x axis, Szz).

have been obtained by means of Method 1 and Method 3,which take these terms into account explicitly, and can beconsidered negligible in both cases.

3.2 MEMS accelerometer

3.2.1 Method 1

Figure 13 represents the sensitivity matrix obtained by ap-plying Method 1, as described in Sect. 2.1, at 3 Hz (upperside) and 6 Hz (bottom side). The relative standard deviation(s_rel), defined in Table 1 and Appendix A, of all main sen-sitivities is negligible (less than 0.02 %).

3.2.2 Method 2

Figure 14 represents the sensitivity matrix obtained by ap-plying Method 2, as described in Sect. 2.2, at 3 Hz (upperside) and 6 Hz (bottom side). Also in this case, the relativestandard deviation (s_rel) appears negligible with respect tothe mean sensitivities, which results in less than 0.09 %.

Figure 13. MEMS accelerometer: sensitivity matrix obtained bymeans of Method 1 (3-D diagram).


3.2.3 Method 3

Figure 15 represents the sensitivity matrix obtained by ap-plying Method 3, as described in Sect. 2.3, at 3 Hz (upperside) and 6 Hz (bottom side). The relative standard devia-tion (s_rel) of all main sensitivities is negligible (less than0.06 %). The offset vectors at f1 = 3 Hz and f2 = 6 Hz areas in the following.qx values are not negligible at both frequencies. For the

MEMS accelerometer under investigation, the digital outputvalues are in metres per second squared (m s−2), and as aconsequence the offset values are expressed in terms of ac-celeration. qxqyqz

f 1

=

0.3576ms−2

−0.06304ms−2

0.1650ms−2

; qxqyqz

f 2

=

0.1662ms−2

−0.07268ms−2

0.1345ms−2

.3.2.4 Comparison between methods

The relative differences between methods (d_rel) obtainedby applying the three methods are reported in Table 4, when




Figure 16. MEMS accelerometer: comparison between methods(main sensitivity along the x axis, Sxx ).

the MEMS sensor is excited at 3 and 6 Hz. Figures 16–18highlight the differences between methods in terms of themean main sensitivities obtained.

Expanded uncertainties (k= 2) of results, estimated ac-cording to D’Emilia et al. (2018), are also represented inFigs. 16–18 by error bars.

The differences of Method 3 with respect to Method 1 arehigher than those of Method 2.

3.2.5 Reproducibility: transverse sensitivities

The relative transverse sensitivities (rel_S) for all axes ob-tained by means of Methods 1 and 3, which take these termsinto account explicitly, are all under 4 %, and can be consid-ered negligible in both cases.

3.3 Discussion of the results

The highest differences of main sensitivity values betweenthe indications of Method 1 and Method 3, of the order of2 %, are associated with higher values of main sensitivity andoffset uncertainty, as evaluated in Method 3. The ability tocalculate together transverse sensitivities and Q= (qi) termsis a valuable capability, even though some interactions be-

Figure 17. MEMS accelerometer: comparison between methods(main sensitivity along the y axis, Syy ).

Figure 18. MEMS accelerometer: comparison between methods(main sensitivity along the x axis, Szz).

tween them could appear, especially when static accelerationcan be measured as in a MEMS accelerometer. No significantdifferences arise in the case of a piezo-electric accelerometer.

The main results are according to the following items:

– the methods offer reproducible results if differences be-tween them and the uncertainty of each method aretaken into account;

– the uncertainty of Method 1, very close to the standardprocedure, is very low, of the order of 0.5 %. Anyway, ifthe calibration procedure is considered, it is more com-plex and expensive than the other ones; furthermore, itseems not suitable for in-field application, with many ofthe procedure steps similar to a primary calibration;

– Method 2 is the quickest and simplest method amongthe ones that have been considered, and it is suitablefor in-field applications. It does not allow us to eval-uate the transversal sensitivities and their effect. Thisaspect strongly affects its uncertainty: if all the uncer-tainty contributions are considered, its relative uncer-tainty is of the order of 5 %. It is unsuitable for applica-



tions where transversal sensitivity effects could be rele-vant.

– The uncertainty of Method 3 is of the order of 3 %, thatis, an intermediate behaviour, as for precision. It is in-tended to be capable of fitting in a satisfactory way mostrequirements for in-line and on-line calibration. Thismethod allows the evaluation of transversal sensitivityand of offset terms; therefore, if these effects are re-markable, they can be estimated and do not affect theprecision of the calibration.

4 Conclusions

The reproducibility of different methods for calibration oftri-axial accelerometers has been evaluated by means of ahigh-performance test bench, based on a linear slide and ahigh-accuracy laser Doppler vibrometer as a reference. Themain sensitivities have been analysed and, where applicable,also transversal ones, in order to get information about thepossibility of using new methods for the on-line and in-linecalibration of MEMS and about the capability of the methodsof satisfying requirements involved in these applications, likelow-cost calibration, operability, traceability and simplicityof procedures.

Two different accelerometers have been tested, a piezo-electric one and a MEMS one of capacitive type, differentwith reference to the ability to measure a constant accelera-tion.

The comparison refers to the following calibration proce-dures:

– Method 1, extending the indications of the standard tothe case of a tri-axial accelerometer;

– Method 2, involving the simultaneous excitation of allthree axes of the accelerometer under test, mounting itonto the surface of a clamp;

– Method 3, describing the relation between the input ac-celerations and the output signals in terms of the sensi-tivity matrix S= (Sij ) and the offset vector Q= (qi).

A low-frequency range of vibration has been studied, 3to 6 Hz, with amplitude of acceleration ranging from 2 to20 ms−2, operated along one axis.

The methods offer reproducible results if differences be-tween them and the uncertainty of each method are takeninto account, for both piezo-electric and MEMS accelerome-ters. Differences and extended uncertainty are of the order ofa few percent.

Methods 2 and 3 appear suitable for in-field applications,even though some differences arise, in particular the follow-ing.

– Method 2 is quick and simple. Anyway, it does not al-low us to evaluate the transversal sensitivities and their

effect. This aspect strongly affects its uncertainty: if allthe uncertainty contributions are considered, its relativeuncertainty is of the order of 5 %. It is unsuitable for ap-plications where transversal sensitivity effects could berelevant.

– The uncertainty of Method 3 is of the order of 3 %, thatis, generally satisfactory, as for precision. This methodallows the evaluation of transversal sensitivity and ofoffset terms; therefore, if these effects are remarkable,they can be estimated and do not affect the precision ofthe calibration. It is able to fit in a satisfactory way mostrequirements for in-line and on-line calibration and tocompensate in a satisfactory manner most of the uncer-tainty causes in the calibration of tri-axial accelerome-ters.

Data availability. Data are not publicly accessible. They are avail-able upon request from the corresponding author.



Appendix A: Parameters used for the comparison

In the following, the parameters of Table A1 have been mademore explicit. The equations referring to Sxx are also appliedto the main sensitivities Syy and Szz and to the offset vector,qx , qy , qz (when Method 3 is taken into account).

It has to be pointed out that the variability of the resultsis evaluated on six entire cycles of oscillation (n= 6 and thesubscript k denotes the kth cycle). For each accelerometer,for each kind of test, three repetitions are executed, in re-peatability conditions.

Table A1. Parameters used for the comparison: symbols and equations.

Quantity Parameter Symbols Equation

Main sensitivity Mean value S, q Sxx =1n∑k

Sxxk

Offset Standard deviation SD SD (Sxx )=

√∑k

(Sxxk−Sxx

)2n−1

Relative standard deviation s_rel s_rel (Sxx )= SD(Sxx )Sxx

· 100

Mean sensitivities of method couples S Sxx12 =Sxx1+Sxx2

2Sxx13 =

Sxx1+Sxx32

Sxx23 =Sxx2+Sxx3

2

Main sensitivity Relative difference between methods d_rel d_rel(Sxx )21 =Sxx2−Sxx1

Sxx12· 100

d_rel(Sxx )31 =Sxx3−Sxx1

Sxx13· 100

d_rel(Sxx )32 =Sxx3−Sxx2

Sxx23· 100

Transverse sensitivity Relative transverse sensitivity rel_S rel_Sxy =Sxy

Sxx· 100; rel_Sxz =

Sxz

Sxx· 100;

rel_Syx =Syx

Syy· 100; rel_Syz =

Syz

Syy· 100;

rel_Szx =Szx

Szz· 100; rel_Szy =

Szy

Szz· 100;



Competing interests. The authors declare that they have noconflict of interest.

Edited by: Nam-Trung NguyenReviewed by: two anonymous referees

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https://doi.org/10.1109/TCST.2010.2076321

https://doi.org/10.1016/j.comcom.2014.09.008

https://doi.org/10.1007/s00542-010-1169-7

https://doi.org/10.1016/j.proeng.2017.05.008

https://doi.org/10.21014/acta_imeko.v4i4.239

https://doi.org/10.1016/j.measurement.2015.08.031

https://doi.org/10.1063/1.4952690

https://doi.org/10.1016/j.measurement.2010.08.016

https://doi.org/10.1088/0957-0233/19/8/085202

https://doi.org/10.1109/TIM.2008.2006137

https://doi.org/10.1109/JSEN.2012.2182991

https://doi.org/10.6028/jres.122.032

https://doi.org/10.1109/I2MTC.2012.6229157

https://doi.org/10.1088/1742-6596/476/1/012119

https://doi.org/10.1088/1742-6596/476/1/012119

https://doi.org/10.1016/j.ndteint.2013.06.003

https://doi.org/10.1109/JSEN.2016.2630008

https://doi.org/10.13140/RG.2.1.1049.4487

https://doi.org/10.1109/WCICA.2002.1021471

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Calibration of tri-axial MEMS accelerometers in the low ... · a capacitive MEMS accelerometer. A low-frequency range of vibration has been investigated, 3 and 6Hz, with amplitude